The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve

Abstract

Let $E$ be an elliptic curve. An irreducible algebraic curve $C$ embedded in $E^g$ is called weak-transverse if it is not contained in any proper algebraic subgroup of $E^g$, and transverse if it is not contained in any translate of such a subgroup.

Suppose $E$ and $C$ are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve $C$ that are close to the union of all algebraic subgroups of $E^g$ of codimension $2$ translated by points in a subgroup $\Gamma$ of $E^g$ of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If $\Gamma$ is trivial, it is sufficient to suppose that $C$ is weak-transverse.

The core of the article is the introduction of a method to determine the finiteness of these sets. From a conjectural lower bound for the normalized height of a transverse curve $C$, we deduce that the sets above are finite. Such a lower bound exists for $g\le 3$.

Concerning the codimension of the algebraic subgroups, our results are best possible.